Measurement of Surface and Interfacial Energies between Solid Materials Using an Elastica Loop

Jia Qi

Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Engineering Mechanics

APPROVED BY:

David A. Dillard, Chairman Raymond H. Plaut

John G. Dillard

September 11, 2000 Blacksburg, Virginia

Keywords: Interfacial Energy, Surface Energy, Work of Adhesion, Contact Mechanics, Elastica, JKR technique

Copyright 2000, Jia Qi

Measurement of Surface and Interfacial Energies between Solid Materials Using an Elastica Loop

(ABSTRACT)

The measurement of the work of adhesion is of significant technical interest in a variety of applications, ranging from a basic understanding of material behavior to the practical aspects associated with making strong, durable adhesive bonds. The objective of this thesis is to investigate a novel technique using an elastica loop to measure the

work of adhesion between solid materials. Considering the range and resolution of the measured parameters, a specially designed apparatus with a precise displacement control system, an analytical balance, an optical system, and a computer control and data acquisition interface is constructed. An elastica loop made of poly(dimethylsiloxane) [PDMS] is attached directly to a stepper motor in the apparatus. To perform the measurement, the loop is brought into contact with various substrates as controlled by the computer interface, and information including the contact patterns, contact lengths, and contact forces is obtained. Experimental results indicate that due to anticlastic bending,

the contact first occurs at the edges of the loop, and then spreads across the width as the displacement continues to increase. The patterns observed show that the loop is eventually flattened in the contact region and the effect of anticlastic bending of the loop is reduced. Compared to the contact diameters observed in the classical JKR tests, the

contact length obtained using this elastica loop technique is, in general, larger, which provides potential for applications of this technique in measuring interfacial energies between solid materials with high moduli. The contact procedure is also simulated to investigate the anticlastic bending effect using finite element analysis with ABAQUS. The numerical simulation is conducted using a special geometrically nonlinear, elastic, contact mechanics algorithm with appropriate displacement increments. Comparisons of the numerical simulation results, experimental data, and the analytical solution are made.

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ACKNOWLEDGEMENTS

I would like to thank Dr. David A. Dillard for acting as my advisor, giving me this opportunity to go deep in this project and providing his guidance throughout my research and writing efforts. I would also like to thank Dr. Raymond H. Plaut of the Civil and Environmental Engineering Department and Dr. John G. Dillard of the Chemistry Department for serving as my committee members.

Additionally, I appreciate the assistance from Prof. M. Chaudhury and his student Hongquan She of Lehigh University, Bob Simonds, as well as all the members in the Adhesion Mechanics Lab. Furthermore, I would like to acknowledge the support of the National Science Foundation under grant CMS-9713949, and the Center for Adhesive and Sealant Science of Virginia Tech.

I would like to take this opportunity to thank my parents for continually supplying me with their love and support throughout my life. Finally, I would like to thank my

husband, Dr. Buo Chen, for his endless supply of love, advice, and understanding.

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Table of Contents

CHAPTER 1 Introduction .......................................................................................... 1 1.1 Adhesion and role of the interface ................................................ 1 1.2 Contact mechanics and measurement of surface energies ............. 2 1.3 Outline of the study.................................................................... 14

CHAPTER 2 Experimental Study............................................................................. 18 2.1 Abstract...................................................................................... 18 2.2 Construction of elastica loop ...................................................... 18 2.3 Sample preparation and material property characterization......... 19

2.3.1 Sample preparation ......................................................... 19 2.3.2 Tensile tests.................................................................... 20 2.3.3 Orientation of the elastica loop ....................................... 21

2.4 Apparatus................................................................................... 22

2.4.1 Displacement control device ........................................... 22

2.4.2 Force measurement device.............................................. 24 2.4.3 Optical system and calibration ........................................ 24 2.4.4 Computer control interface and data acquisition system.. 25

2.5 Measurement using the JKR method .......................................... 25 2.6 Measurement using elastica loop ................................................ 27

2.6.1 Contact patterns.............................................................. 28 2.6.2 Experimental results and discussion................................ 29 2.6.3 Hysteresis ....................................................................... 32

CHAPTER 3 Numerical Analysis............................................................................. 65 3.1 Abstract...................................................................................... 65 3.2 Meshes and boundary conditions................................................ 65 3.3 Simulation of the loop formation................................................ 66 3.4 Contact simulation ..................................................................... 67

3.4.1 Contact principle ............................................................ 67

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3.4.2 Contact procedure........................................................... 70 3.4.3 Contact patterns.............................................................. 70 3.4.4 Strain of the elastica loop after contact............................ 71

3.4.5 Contact force and contact area ........................................ 71 3.4.6 Poisson's ratio effects...................................................... 72

CHAPTER 4 Comparison of Experimental, Numerical, and Analytical Results........ 93 4.1 Analytical solution ..................................................................... 93 4.2 Comparison of numerical and analytical results.......................... 95 4.3 Comparison of experimental and analytical results ..................... 97 4.4 Sensitivity studies of errors ........................................................ 99 4.5 PDMS loop in contact with various substrates .......................... 101

CHAPTER 5 Conclusions and Recommendations for Future Work........................ 121 5.1 Summary and conclusions ........................................................ 121 5.2 Limitations............................................................................... 123 5.3 Future work.............................................................................. 124

References ......................................................................................................... 126 Appendix A. Mathematica file for fitting the work of adhesion using traditional JKR

testing technique. .............................................................................. 132 Appendix B. ABAQUS input file for contact simulation........................................ 133

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List of Figures

Figure 1.1 The relationship between the work of adhesion and peel strength. After reference 6. ................................................................................... 15

Figure 1.2 Contact angle measurement for a liquid drop on a solid surface and surrounded by a vapor............................................................................ 16

Figure 1.3 Interfacial energy's effect on the contact zone: the solid lines show the actual contact situation with contact radius a1, and the dashed lines show the Hertzian contact solution with contact radius a0. After reference 12. .......................................................................................... 17

Figure 2.1 Elastica loop configuration .................................................................... 35 Figure 2.2 A typical stress-strain plot for a PDMS strip. ......................................... 36 Figure 2.3 Youngs moduli for eight PDMS samples of the first batch and their

average. The error bar represents one standard deviation..................... 37

Figure 2.4 Youngs moduli for four PDMS samples of the second batch and their

average. The error bar represents one standard deviation..................... 38

Figure 2.5 Youngs moduli for four PDMS samples of the third batch and their

average. The error bar represents one standard deviation..................... 39

Figure 2.6 Geometry of an elatica loop probe ......................................................... 40 Figure 2.7 Schematic of apparatus .......................................................................... 41 Figure 2.8 Photograph of the apparatus................................................................... 42 Figure 2.9 Front panel of LabVIEW VI created for the computer acquisition

system. .................................................................................................. 43

Figure 2.10 Interaction between a PDMS lens in contact with a PDMS film coated on a glass substrate. The dimensions of the figure are not to scale. ........ 44

Figure 2.11 Typical contact pattern of a PDMS lens in contact with a PDMS film coated on a glass substrate. .................................................................... 45

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Figure 2.12 Plot of contact radius, a, vs. force, F, in JKR testing: a PDMS lens in contact with a PDMS film coated on a glass substrate. ........................... 46

Figure 2.13 Plot of cube of contact radius vs. contact force: the dots represent

loading and unloading experimental data and the solid lines represent fitting curves obtained using a numerical regression method.................. 47

Figure 2.14 Interaction between a PDMS loop in contact with a PDMS film coated on a glass substrate. ............................................................................... 48

Figure 2.15 Anticlastic bending of an elastica loop. .................................................. 49 Figure 2.16 Initial contact zone of a PDMS elastic loop in contact with a PDMS

coating on a glass substrate, and about two-thirds of the loop width is shown. ................................................................................................... 50

Figure 2.17 Further contact zone of a PDMS elastic loop in contact with a PDMS coating on a glass substrate, but the contact length is smaller than one third of the elastica loop width. .............................................................. 51

Figure 2.18 A typical contact zone of a PDMS elastica loop in contact with a

PDMS coating on a glass substrate......................................................... 52 Figure 2.19 Plot of contact length, 2B vs. contact force, F: a PDMS loop in contact

with PDMS coating on a glass plate. The solid lines represent the fits of data within the two different regions, which indicate that each

region has a different contact spreading rate........................................... 53 Figure 2.20 Plot of contact force vs. the displacement of the loop: PDMS loop in

contact with PDMS substrate. ................................................................ 54 Figure 2.21 Plot of contact length, 2B vs. contact force, F: PDMS loop in contact

with a glass plate. The solid lines represent the fits of data within the

two different regions, which indicate that each region has a different contact spreading rate. ........................................................................... 55

Figure 2.22 Plot of contact force vs. displacement of the loop: PDMS loop in contact with a glass plate. ...................................................................... 56

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Figure 2.23 Plot of contact length, 2B vs. contact force, F: PDMS loop in contact with a PC plate. The solid lines represent the fits of data within the two different regions, which indicate that each region has a different

contact spreading rate. ........................................................................... 57 Figure 2.24 Plot of contact force vs. the displacement of the loop: PDMS loop in

contact with a PC plate. ......................................................................... 58 Figure 2.25 Plot of contact length, 2B, vs. contact force, F: PDMS loop in contact

with backing of commercial cellulose acetate substrate (3M ScotchTM Transparent Tape 600). The solid lines represent the fits of data within the two different regions, which indicate that each region has a

different contact spreading rate. ............................................................. 59 Figure 2.26 Plot of contact force vs. the displacement of the loop: PDMS loop in

contact with backing of commercial cellulose acetate substrate (3M ScotchTM Transparent Tape 600). ......................................................... 60

Figure 2.27 DMA creep test results using a rectangular PDMS strip at room temperature with a 0.2 MPa stress.......................................................... 61

Figure 2.28 Room temperature stress-strain curves of PDMS for three loading-

unloading cycles. The loading and unloading rate used in the tests was 5mm/min. .............................................................................................. 62

Figure 2.29 Plots of contact length, 2B vs. contact force, F for three loading-unloading cycles: PDMS loop in contact with PDMS substrate. ............. 63

Figure 2.30 Plots of contact length, 2B, vs. contact force, F, for two constant loading-unloading rates with zero dwell intervals: PDMS loop in contact with PDMS substrate. ................................................................ 64

Figure 3.1 FEA mesh of elastica loop with dimensions 14.71 mm 0.956 mm

0.165 mm, modulus 1.81 MPa, and a series of Poissons ratios of 0, 0.1, 0.2, 0.3, 0.4, 0.49. ........................................................................... 75

Figure 3.2 Simulating formation of the elastica loop. .............................................. 76 Figure 3.3 Longitudinal strains on the concave side of the loop before contact........ 77

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Figure 3.4 Transverse strains of the loop on the concave side before contact........... 78 Figure 3.5 Contact schemes. ................................................................................... 79 Figure 3.6 Contact actions simulated in FEA. ......................................................... 80 Figure 3.7 Contours of displacement of initial contact at edges with insert

showing corresponding experimental results. ......................................... 81 Figure 3.8 Compressive stress distribution on the upper loop suface of initial

contact at edges...................................................................................... 82

Figure 3.9 Contours of displacement after contact area spreading with insert showing corresponding experimental result............................................ 83

Figure 3.10 Contours of contact pressure on the upper loop surface after contact area spreading........................................................................................ 84

Figure 3.11 Longitudinal strain on the concave side of the loop after contact. ........... 85 Figure 3.12 Transverse strain on the concave side of the loop after contact............... 86 Figure 3.13 Plot of contact length vs. contact force: the contact lengths were

measured at the center point of the contact front. ................................... 87

Figure 3.14 The effects of the measurement position on contact length: the

dimensions of the elastica strip are 14.7 mm 0.96 mm 0.17 mm,

E= 1.81 MPa, = 0.49, and 2C = 6.38 mm. ........................................... 88

Figure 3.15 Effects of Poissons ratio on the contact patterns with full width of the modeled elastica loop............................................................................. 89

Figure 3.16 The relationship between contact length and contact force. The contact length was taken as the center of the contact front...................... 90

Figure 3.17 The relationship between contact length and the contact force. The contact length was taken as the point with maximum pressure. .............. 91

Figure 3.18 The relationship between contact length and contact force. The contact length was taken as the outer edge of the contact front. .............. 92

Figure 4.1 An elastica loop in contact with a flat, rigid, smooth, horizontal

surface. After reference 46. ................................................................. 104 Figure 4.2 Definitions of variables. After reference 46......................................... 105

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Figure 4.3 Model adhesion effects. After reference 46. ........................................ 106 Figure 4.4 Comparison of numerical and analytical results: contact lengths were

measured at different positions............................................................. 107

Figure 4.5 Comparison of numerical and analytical results: contact lengths were measured in the cases with different value of Poissons ratio at the center of contact front. ......................................................................... 108

Figure 4.6 Comparison of numerical and analytical results: contact lengths were measured in the cases with different values of Poissons ratio at the points of maximum pressure. ............................................................... 109

Figure 4.7 Comparison of numerical and analytical results: contact lengths were measured in the cases with different values of Poissons ratio at the

outer edge of contact front. .................................................................. 110 Figure 4.8 Comparison of the experimental and analytical results: PDMS loop in

contact with PDMS film coated on glass plate...................................... 111 Figure 4.9 Comparison of the experimental and analytical results: PDMS loop in

contact with a glass plate. .................................................................... 112

Figure 4.10 Comparison of the experimental and analytical results: the PDMS loop in contact with cellulose acetate substrate (3M ScotchTM Transparent Tape 600). ........................................................................................... 113

Figure 4.11 Comparison of analytical, experimental and numerical results: PDMS loop in contact with PDMS film coated on a glass plate, and the contact lengths were taken from the center of contact front. ................. 114

Figure 4.12 Effects of reducing length by 5%, 10%, 20%. ...................................... 115 Figure 4.13 Effects of increasing width by 10%, 20%, and 30%. ............................ 116 Figure 4.14 Effects of increasing thickness by 5%, 10%, and 15%.......................... 117 Figure 4.15 Effects of reducing the separation distance between the ends by 5%,

10%, and 20%...................................................................................... 118 Figure 4.16 Effects of increasing Youngs modulus by 10%, 20%, and 30%........... 119

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Figure 4.17 Plots of contact length, 2B vs. contact force, F: PDMS loop in contact with four different substrates................................................................ 120

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List of Tables

Table 2.1 Youngs modulus of the tensile specimens from different batches.......... 34 Table 4.1 Nondimensionalization process............................................................ 103

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CHAPTER 1 Introduction

1.1 Adhesion and role of the interface

Adhesion is a phenomenon by which two materials in contact form a region of adhesive bond that is able to sustain and transmit stresses. The formation of adhesive bonds is governed by molecular interactions occurring at the interface of the adhering

materials. As discussed in Kinloch [1], scientists have recognized for many years that intimate molecular contact and active interactions are necessary, though sometimes insufficient, requirements to form strong adhesive bonds. As summarized by Mangipudi and Tirrell [2], these interactions include: i) van der Waals or other non-covalent interactions that form bonds across the interface; ii) interdiffusion of polymer chains across the interface and coupling of the interfacial chains with the bulk polymer; and iii) formation of primary chemical bonds between chains or molecules at or across the interface.

The van der Waals and other non-covalent interactions at the interface are macroscopic intrinsic material characteristics, and they are quantified as the surface and interfacial energies. Defined as the energy required to create a unit area of surface of a

material in a thermodynamically reversible manner, the surface energy of a material, ,

and the interfacial energy between two materials in contact, 12, determine the work of cohesion, Wcoh, for two identical surfaces, or the work of adhesion, Wadh, for two

dissimilar surfaces in contact [3]. For two identical surfaces in contact, Wcoh is given by 2=cohW (1)

For two dissimilar surfaces in contact, Wadh is given by

1221 +=adhW (2) where 1 and 2 are the surface energies of materials 1 and 2, respectively, 12 is the interfacial energy between materials 1 and 2.

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Although the practical fracture toughness of an adhesive bond, which is usually quantified as the critical strain energy release rate Gc, is in general several orders of magnitude larger than the thermodynamic work of adhesion Wadh, sufficient evidence in

the literature has shown that the strength or fracture toughness of adhesive bonds is directly associated with the work of adhesion [[4], [5], [6]]. As indicated in Figure 1.1, Wightman et al. [6] showed that the peel strength of a pressure sensitive tape increases significantly as the work of adhesion increases. Other examples can be found in

references [4] and [5]. Gent and Schultz [4] proposed that the fracture toughness, Gc, and work of adhesion, Wadh, follow the relationship

( )[ ]TvWG adhc ,1 += (3) where represents the viscoelastic energy dissipation associated with the debonding

process, and is a function of the debonding rate v and testing temperature T.

Although equation (3) is obtained directly based on experimental observations, the equation underlines the importance of the thermodynamic work of adhesion to a

strong and durable adhesive joint. Consequently, measurement of surface and interfacial energies is of significant technical importance in aspects such as obtaining a basic understanding of material behavior and manufacturing durable adhesive bonds.

1.2 Contact mechanics and measurement of surface energies

The measurement of surface energies of solids has been a difficult task [7]. This is due to the fact that the moduli of solids in general are relatively high (as compared to liquids and vapors), and the deformation due to the surface energy is usually insufficient to be measured. Historically, estimates of the surface energies of solids have been obtained through measuring the surface energy of the melt and assuming that the energetics of the solid and liquid are similar. However, since the surface energy of a

material is closely related to the surface temperature, results obtained using this melting technique contain significant errors [3]. Moreover, this technique is very difficult to

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apply to high-molecular-weight polymers, which are very viscous even in the melt. In

addition, this technique cannot be applied to cross-linked polymers, because cross-linked polymers cannot be melted.

Currently, the most widely used technique to estimate surface energies of solids is the wetting method [3]. The basis of this technique is Youngs wetting equation, which is given by

slsvlv +=cos (4) where, as shown in Figure 1.2, subscripts s, l, and v denote the solid, liquid, and vapor

phases, and lv, sv, and sl are the interfacial energies between the various phases. In practice, a serie of liquid droplets with different liquid-vapor interfacial energies is used

as probes to measure contact angles with a solid surface. If the solid-liquid interfacial

energy is assumed to be small, an estimate of the solid-vapor interfacial energy sv can then be obtained by extrapolating the results to = 0, which is also referred as the critical

surface tension c. This wetting procedure was first introduced by Zisman, and is relatively easy to perform [3]. However, the solid-liquid interactions are essentially ignored in the method, which reduces the accuracy of the results significantly in some cases. More importantly, especially for the purpose of adhesion, this technique cannot be

extended to the measurement of interfacial energies between two solids.

Another useful quantity for characterizing the interactions between two materials is the work of adhesion, Wadh,

[3]. After the contact angle of a droplet is measured, the work of adhesion between the substrate and the liquid of the droplet can be obtained as

( ) cos1+= lvadhW (4) By knowing the work of adhesion between various solids and liquids, interactions between two solids can be estimated. However, some critical assumptions are often required and consequently errors are introduced, which in some cases may not be negligible.

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Of significant recent interest has been the use of contact mechanics to study interfacial interactions. The seminal study of contact mechanics was conducted by Hertz

[8], who analyzed the deflection, contact area, and stress distributions of two elastic spheres in contact, and obtained relationships between the contact radius, contact force, and deflection as

KPR

a =30 (5)

Ra

KRP 20

2

== (6)

21

111RRR

+= (7)

+

=

2

22

1

21 11

431

EEK

(8)

In equations (5)-(8), as shown in Figure 1.3, a0 is the contact radius, is the deflection, P is the contact force, and R1, R2, E1, E2, 1, and 2 are the radii, moduli, and Poisson's

ratios for the two spheres in contact, respectively. In Hertz's theory, the materials are

linearly elastic; contact surfaces are frictionless; and the interfacial attractive forces between the spheres are ignored. Consequently, if the contact force decreases to zero,

this theory predicts that the contact area is also zero as indicated in equation (5). Lee and Radok [9], Graham [10], and Yang [11] later investigated the contact problem between viscoelastic solids. However, none of these treatments accounts for the influence of the interfacial interactions.

Johnson, Kendall, and Roberts [12] showed that if one of the spheres in contact is elastomeric, the contact area measured is larger than what Hertz predicted. In addition, they also showed that the contact area at zero contact loads has a finite size and requires a small tensile force to separate the spheres. They recognized that this additional

deformation is due to the interfacial energy between these two spheres, and extended

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Hertzian theory to include the influence of surface energies in their analysis, which is now known as the JKR theory.

In developing the JKR theory, Johnson et al. [12] considered two similar, solid, homogeneous, linearly elastic spheres with frictionless surfaces in contact with each

other under a normal applied load P as shown in Figure 1.3. The shear tractions acting on the contact surfaces are identically zero. The deformation of the spheres is assumed

to be small so that linear continuum mechanics theory can be used. In particular, small strains are required so that there is no distinction between the deformed and undeformed configurations of the bodies as far as equilibrium is concerned. In addition, the bonding and debonding process at the interface is assumed to be reversible and the energy

available for debonding is independent of the local failure process.

According to the JKR theory, the contact radius, a, resulting from the combined influences of the interfacial interactions and the external force, P, is given by

( ) ]36[3 23 RWRPWRWPKR

a adhadhadh pipipi +++= (9)

where Wadh is the work of adhesion between the two spheres and the rest of the symbols

are as stated before. Based on equation (9), the work of adhesion Wadh can be calculated from knowledge of the contact radius, the applied load, and the material properties of the spheres. Equation (9) also indicates that due to the interfacial interactions, the resultant contact radius a is larger than the contact radius a0 predicted by equation (5), which represents the Hertzian theory. In addition, equation (9) also predicts that when the contact force is zero, the contact area will not be zero, and a tensile load is required to separate the two spheres.

The JKR theory provided a new experimental technique to directly measure the work of adhesion between two solids and consequently the surface and interfacial energies. A variety of experimental tools such as the JKR apparatus and the surface

6

force apparatus (SFA) also have been developed. All of the apparatuses are based on the JKR theory and measure the same quantities, i.e., the contact area (by means of contact radius or contact length) and the applied load. Using these tools, many research investigations in various areas have been conducted.

Tabor [13] examined the effects of surface roughness and material ductility on the adhesion of solids. The results indicated that the adhesion between two surfaces

decreases as the roughness of the surface increases. However, the rate of decrease depends on the moduli of the materials in contact. Chaudhury and Whitesides [14] tested the surface energies of poly(dimethylsiloxane) [PDMS] in air and in mixtures of water and methanol. The results showed that the interfacial interactions decrease in the

mixtures of water and methanol as the methanol content increases. However, a small interaction persists, even in pure methanol. In both the Tabor [13] and Chaudhury and Whitesides [14] experiments, the probes used were hemispherical elastomeric lenses.

After the success in measuring the interfacial energy between elastomeric materials, efforts have been made to measure the interfacial energy between glassy polymers. Using their surface force apparatus (SFA), Mangipudi et al. [15] obtained the thermodynamic work of cohesion and adhesion between poly(ethylene terephthalate) [PET] and polyethylene thin films. The surface energy of PET determined by the direct force measurement is higher than the critical surface tension of wetting. They later

applied the surface force apparatus (SFA) to measure the surface energies of polyethylene films modified with corona treatment [16]. Comparing the results with those obtained using the contact angle measurement technique, they also believed that

the contact angle measurement technique is not sensitive to small changes in surface

composition, and SFA can directly measure the true surface energy of polymer films.

Tirrell [17] discussed the applications of JKR theory for glassy or semi-crystalline polymers. In his experiments, high modulus materials were coated on the

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elastomer lenses and contacted with flat surfaces coated with the same polymer. He also compared the results with those obtained using contact angle measurements, and obtained similar conclusions as Mangipudi et al [15] that the results obtained using the contact JKR method are larger than those obtained using the contact angle technique. Tirrell [17] attributed the difference to the rearrangement of the polymer functionality group on the surface. For polar materials, high-energy functional groups on the surface

may be buried when in contact with air, but might rearrange to expose themselves when

in contact with a polar polymer surface.

Mangipudi et al. [18] further expanded the applicability of the classical JKR experiment to allow measurements of the work of adhesion between glassy polymers

using a novel method to prepare samples, and they measured the surface energies of polystyrene [PS] and poly(methyl methacrylate) [PMMA]. In these experiments, the spherical cap was made of O2-plasma modified cross-linked PDMS, and the glassy polymer was coated on the PDMS cap as a thin layer about 0.1 m thick by solvent

casting. Detailed experimental procedures are presented in their paper. During their experiments, contact hysteresis was observed in PMMA-PMMA and PS-PS interfaces. Similar to Tirrell [17], they also concluded that contact hysteresis is due to contact induced rearrangements of the interface since no contact hysteresis was observed in

PDMS-PS and PDMS-PMMA interfaces. Ahn and Shull [19] investigated the work of adhesion between a lightly cross-linked poly(n-butyl acrylate) [PNBA] hemispherical lens and a PMMA flat surface, and contact hysteresis was observed at all accessible rates of unloading. More contact hysteresis work will be discussed later in this section. Other work worth mentioning is Shull et al. [20], who investigated nonlinear elasticity effects and extended the small contact radius assumption used in the JKR theory by using a correction factor.

Besides the spherical geometry, other sample geometries have also been used to

study interfacial interactions. Using SFA and adhering thin mica sheets to two

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cylindrical glass lenses, Horn et al. [21] investigated the contact between two solids. In their study, the axes of the two cylinders were perpendicular to each other, and the separation distance between the mica layers was measured using an optical

interferometer through a layer of silver coating between the mica sheets and the glass lenses. The SFA apparatus was either filled with a KCL solution or N2 during the experiments in order to achieve different interfacial interactions. For the tests involving KCL solution, which represents non-adhesive contact, the contact radius increases with

the load following the Hertzian theory. For the tests involving N2, which represents the adhesive contact, results indicated that a greater contact area is observed under the same load, and a finite contact area exists under zero load conditions that can be analyzed using the JKR theory. However, due to the layered structure, Sridhar et al. [22] pointed out that there will be some errors using JKR theory to analyze the experiment, and they developed an extended JKR theory to analyze the layered structure by introducing an

adhesion parameter 31

*

1

22hERWadh

= , layer thickness ratio, and the ratio of elastic moduli

to characterize the interfacial interactions. In their paper, Sridhar et al. [22] also conducted a finite element analysis using ABAQUS to simulate the contact procedure and verify their analytical model.

In JKR theory, the interfacial forces are assumed to act only within the contact

region, and consequently a stress singularity results at the contact edge. Derjaguin et al. [23] suggested that the attractive forces between two solids also exist in a small zone just outside of the contact region, and developed a new theory. In their analysis, deformation was assumed to follow Hertzian theory, and the attractive forces were modeled as van

der Waals' force. However, Muller et al. [24] showed that more accuracy would be gained by using a Lennard-Jones potential to model the attractive forces. Errors in Derjaguin et al. [23] were corrected in Muller et al. [25] and Pashley [26], in which equilibrium conditions were obtained by balancing the elastic reaction forces with the

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surface forces and the applied load. Due to the efforts of these researchers, this theory is now known as the DMT theory.

Because different approaches had been used to develop the theories, and results predicted were significantly different [27], there was controversy between the JKR and the DMT theories. However, the controversy has been satisfactorily resolved. According to Tabor [28], JKR theory should be used when the dimensionless parameter

3/1

20

2

2

=

zKRWadh (10)

is greater than five. In equation (10), R and K are as defined earlier in this chapter, Wadh is the work of adhesion, and z0 is the equilibrium separation distance of the contact

surface. Physically, the DMT theory is more appropriate to use for high modulus solids with low surface energy and small radius of curvature. On the other hand, JKR theory should be used for low modulus solids with high surface energy and large radius of curvature.

Using the Dugdale model for fracture mechanics, Maugis [29] treated the contact edge as a propagating crack in mode I and developed a general theory to form a link between the DMT and JKR theories. Similar to Tabor [28], the results again showed that the JKR theory is valid only for short-range interactions and/or for soft materials; in

contrast, the DMT theory is applicable for long-range interactions and/or for hard materials. Maugis and Gauthier [30] showed a JKR-DMT transition in the presence of a liquid meniscus around the contact using a surface force apparatus. Baney and Hui [31] extended Maugiss work. They modeled a cohesive zone to describe the adhesion

between long cylinders in contact and found the valid regions for the JKR and DMT theories, respectively.

As discussed earlier in this thesis, materials are assumed to be linearly elastic,

isotropic, and homogeneous in the JKR theory, and the bonding and debonding processes

10

are assumed to be reversible. While this is true for some model systems, most systems exhibit some irreversibility. As a result, hysteresis appears during the loading and unloading process as discussed in Mangipudi et al. [18] and Ahn and Shull [19]. Due to the contact hysteresis, a lower compressive force is needed to obtain a given contact radius during the unloading phase than is required during the loading phase. Silerzan et al. [32] demonstrated that there is a large hysteresis between the loading and unloading regimes of siloxane elastomers, and proposed a generalized JKR model to explain the

experimental data. Kim and his co-workers [33], [34] studied the adhesion and adhesion hysteresis using cross-linked PDMS hemispherical surfaces and a self-assembled model surface containing different chemical functional groups. Using the JKR method, they observed that the hysteresis resulting from a fast relaxation process is practically

eliminated using stepwise loading and unloading protocols. The contact pressure-induced, interfacial hydrogen bonds were shown to make significant contributions to the contact hysteresis and were used to explain the phenomena observed. She et al. [35] used a rolling contact geometry to study the effects of dispersion forces and specific

interactions on interfacial adhesion hysteresis. Hemi-cylindrical elastomers (both unmodified and plasma oxidized) were rolled on PDMS thin films bonded to silicon wafers. The results indicated that the adhesion hysteresis in the PDMS-plasma oxidized PDMS system depends significantly on the molecular weight of the grafted polymer,

whereas the hysteresis is rather negligible for the pure unmodified PDMS system. These results are explained in terms of hydrogen bonding, and orientation and relaxation of polymer chains. Li and Tirrell [36] measured the surface energy of an acrylic pressure sensitive adhesive PSA-LN-NoAA and the hysteresis behavior under different loading/unloading speeds. They showed that 27% adhesion hysteresis exists between the

equilibrium values from the curves of contact radius versus force in the loading and unloading procedures.

To investigate the work of adhesion, JKR theory has also been applied to the

atomic force microscope (AFM), which provides a higher accuracy for the contact radius

11

and load measurement. Carpick et al. [37] used a platinum coated AFM tip in contact with the surface of mica in an ultrahigh vacuum, and determined the interfacial adhesion energy and shear stresses using the JKR theory. Gracias and Somorjai [38] modified the AFM using a tip with a large radius of curvature to reduce the pressure in the contact region. They then measured the surface energies of low-density polyethylene, high-density polyethylene, isotactic polypropylene, and atactic polypropylene. K. Takahashi et al. [39] studied the stiffness of the measurement system such as the AFM cantilever and the significant figures of measured displacement. They again confirmed the transition between JKR theory and DMT theory discussed by Maugis [29].

Contact mechanics has also been substantially used in investigations for the

adhesion of particles to various substrates. Rimai et al. [40] measured the surface force-induced contact radii of 20 m glass particles deposited on polyurethane substrates with

a 5 m thick thermoplastic layer. The results showed that the size of the deformations depends only on the coating and not on the modulus of the underlying substrate. Soltani et al. [41] studied the particle removal mechanism from rough surfaces due to an accelerating substrate. The rough surface was modeled by asperities of the same radius of curvature and with heights following a Gaussian distribution. The JKR theory and the

theory of critical moment, sliding, and detachment were used to analyze particle pull-off forces, and to evaluate the critical substrate accelerations for particle removal. Reasonable agreement between the model prediction and experimental data was obtained for aluminum and glass particles. Gady et al. [42] measured the force needed to remove micrometer-size polystyrene particles from elastomeric substrates with Youngs moduli of 3.8 and 320 MPa using an atomic force technique. They found that the removal force differed by an order of magnitude between the two substrates. When the more compliant substrate was overcoated with a thin layer of the more rigid material, the removal force increased with increasing applied load, which is in conflict with the predictions by the

JKR theory and can be explained by taking into account the roughness of the particle and the amount of embedment of the particle into the substrate.

12

In brief, contact mechanics technology has brought great success in the measurement of interfacial interactions. Recently, Mangipudi and Tirrell [2] gave a complete review of the recent developments in the theories of contact mechanics, and their applications in the design and interpretation of experimental measurement of

molecular level adhesion between elastomers, glassy polymers and viscoelastic polymers. They also identified some potential applications in other fields, such as biomaterials.

Despite all the successes achieved so far in the measurement of surface and interfacial energies, techniques presently used are only applicable to certain types of materials. For example, the JKR apparatus requires a compliant, elastomeric lens.

However, not all solid polymers of interest for interfacial energy measurement have bulk mechanical properties amenable to the JKR-type of analysis. In particular, many

materials have a modulus higher than is convenient to measure a significant increase in contact radius in the linear contact mechanics regime. If the hemispherical lens is directly made of a stiffer material, such as an epoxy, the change of contact zone due to interfacial interactions is too small to be measured easily. Liechti et al. [43] reported pioneering investigations measuring the work of adhesion of the epoxy/glass interface and compared it with the work of adhesion obtained from contact angle measurement. If interfacial energies between two solid materials, neither of which is an elastomer, are desired, the lens must be coated with one of the materials of interest, which can be an

inconvenience. In addition, not all materials can be deposited on such a lens. As an example, present testing methods cannot be used to determine the interfacial energy between aluminum and steel. Consequently, efforts have also been made to investigate other testing technologies, by which the surface energies for a broader range of materials

can be measured directly.

13

Recently, Shanahan [44] published a mathematical model to measure interfacial energies using a spherical membrane or "balloon". In his model, a spherical membrane under slight internal pressure is brought into contact with a flat and rigid surface. The

results of his analysis showed that the interfacial energy depends on the square of the contact radius rather than the cube of the contact radius as with the JKR theory. As a result, the potential inaccuracy induced from the measurement error of the contact radius can be significantly reduced. Additionally, Shanahan [44] also showed that the contact area observed using the balloon method is approximately 10 times larger than that observed in JKR tests under the same load. This greater contact area also provides greater convenience for the test. However, in practice, sample preparation for this

method is very difficult, and no experimental data has yet been published.

Plaut et al. [45] modeled an elastica loop with a fixed separation distance pushed into contact with a flat surface. In their analysis, the surface energies of the materials

were ignored, and the elastica was assumed to be thin, uniform, smooth, inextensible,

and flexible in bending. Through the analysis, the conformations of an elastica loop for a given displacement of the loop as well as the contact length and the resultant contact force were obtained. Because the structure of the elastica loop is very compliant even though the modulus of the material is high, measurable deformations due to interfacial

interactions when the loop is in contact with another surface may be obtained. The analysis of Plaut et al. [45] provides a potential method to measure the surface and interfacial energies between arbitrary material systems. In conjunction with this thesis, Dalrymple [46] extended the solution of Plaut et al. [45] and included surface energies in the analysis. The study [46] serves as the analytical background of this study.

In this study, a novel technique using soft structures instead of soft materials (i.e. elastomers) is proposed and investigated to measure surface and interfacial energies between solids. In practice, a probe is first constructed as a thin, uniform, smooth, and

flexible-in-bending elastica loop using the material of interest. Then the elastica loop is

14

brought into contact with a flat rigid surface made of another material of interest. Because of the compliant structure, measurable changes in the deformation of the elastica due to interfacial interactions are obtained. Since the final conformation of the elastica is

the result of the applied load and the interfacial attractive forces, the interfacial energy can be estimated by analyzing the relationship between the contact area and the applied load of the loop.

1.3 Outline of the study

This thesis is divided into five chapters. Chapter 1 gives the project background, the literature review, the objective of this research, and the outline of the study. Chapter 2 discusses the development of the testing method, the experimental procedure, and the results. Chapter 3 shows the numerical analysis of the contact procedure of the elastica loop in contact with a rigid flat surface using ABAQUS 5.8 [47]. The comparison and discussion among the experimental results, numerical simulation, and analytical results are presented in Chapter 4. Finally, in Chapter 5, the major conclusions are summarized as well as some suggestions related to this research work.

15

Figure 1.1 The relationship between the work of adhesion and peel

strength. After reference 6.

400

200

0 40 60 80 100

0

0.04

0.08

0.12

Peel

Stre

ngth

(J/m

2 )

Wor

k of

Adh

esio

n (J

/m2 )

Si / (Si + S) (Atomic %)

16

Figure 1.2 Contact angle measurement for a liquid drop on a solid surface and surrounded by a vapor.

S

L

V

sl

sv

lv

lv

sv

sl

S

17

Figure 1.3 Interfacial energy's effect on the contact zone: the solid lines

show the actual contact situation with contact radius a1, and the dashed lines show the Hertzian contact solution with contact radius a0. After reference 12.

3

a 0

a 1

a 1

a 0

R 1

R 2

P

5

5

3

18

CHAPTER 2 Experimental Study

2.1 Abstract

In the attempt to develop a method that can be used to measure surface and interfacial energies of solids for a broader range of material types, the use of a flexible structure has been proposed. An elastica loop was chosen for use in this research. In the

apparatus, the elastica loop is directly attached to the shaft of a stepper motor that is controlled by a computer. When the elastica loop is brought into contact with a flat substrate, the interfacial attractive force produces a measurable change in contact area. The contact pattern is observed in the monitor through a Nikon macro lens with a

magnification factor of 50x, and the contact length is measured. At the same time, the force F between the loop and flat substrate is also measured using an analytical balance.

Distinct contact patterns of the elastica loop made of poly(dimethylsiloxane) [PDMS] in contact with a variety of substrates are observed, and the effect of anticlastic bending is eliminated in the center of the contact area due to the flattening of the loop. As compared to the classical JKR tests, the force applied is smaller, the contact length is larger, and the displacement of the loop applied is also larger. Large contact hysteresis

with a tail due to interfacial interactions has also been observed in the tests.

2.2 Construction of elastica loop

To directly measure surface and interfacial energies of solids for a broader range of materials compared to the JKR technique, an elastica loop probe with characteristics of a compliant structure is prepared using the material of interest. To achieve the desired structural compliance, the elastica strip is prepared to be thin, uniform, smooth, and

flexible in bending. During the test, the elastica loop is brought into contact with a rigid, flat surface made of, or coated with another material of interest, and the compliant

19

structure permits measurable deformation due to small interfacial attractions. Through the analysis of the deformation, the interfacial energy between the two materials can be obtained. The analytical solution for an elastica in contact with a flat surface is given in

Dalrymples thesis [46].

The dimensions of the elastica and the forces acting on the system are defined as follows. The total length of the elastica strip is 2L, the width of the strip is W, the

thickness is t, the bending stiffness is EI, and the distance between the two ends of the strip is 2C. The ends of the elastica are first lifted, bent, and then clamped vertically at an equal height with a specified distance apart. When the bent elastica strip is brought into contact with a flat substrate, the contact length is a result of the applied load and the

interfacial attractive force. The total the contact length is defined as 2B, and the resulting reaction force is indicated as F in Figure 2.1.

2.3 Sample preparation and material property characterization

2.3.1 Sample preparation

In this study, the major material used for the elastica loop is SYLGARD 184 silicone elastomer (PDMS) provided by the Dow Corning Company because this material is very stable over a wide temperature range (-50C to 200C), and has a very low water absorption and very good radiation resistance. The choice of an elastomeric material for the elastica loop allows comparison of the results with findings obtained

using the JKR method. This comparison is a necessary step to verify the methodology. The SYLGARD 184 silicone elastomer contains a silicone base and curing agents and is supplied in a two-part kit comprised of liquid components. The base and the curing agent were mixed in a ratio of 10 parts base to 1 part curing agent by weight with gentle

stirring for about 10 minutes to minimize the amount of air introduced. The mixture then rested in air for 30 minutes to remove the air bubbles before use. The PDMS liquid

20

mixture was then poured on a glass plate cleaned with acetone, and a doctor's blade was used to spread the liquid and to control the thickness of the film to be 1.0 mm.

SYLGARD 184 silicone elastomer offers a flexible cure temperature from 25C to

150C for various amounts of time, and requires no post-cure. In this study, after the

PDMS liquid mixture was spread on the glass surface, the plate was then placed in a programmable oven. The curing process started from room temperature and the

temperature was increased at a rate of 5C/min until 100C, where the temperature was maintained for 1 hour. Then the temperature was decreased to room temperature at a rate of 5C/min. After curing, the film was peeled off from the glass plate. The product was a homogeneous, transparent, and flexible film with a final thickness about 0.2 mm, which

varied slightly from batch to batch. Specimens were then cut from the film with appropriate dimensions for various mechanical tests.

Various substrates were selected to perform the measurement of surface energies.

They were glass plates coated with PDMS, acetone-washed glass plates, polycarbonate

[PC] plates, and a commercial cellulose acetate substrate (3M ScotchTM Transparent Tape 600). The preparation of the substrates was relatively straightforward and after preparation, all the substrates, as well as the elastomer films, were stored in a desiccator at room temperature with relative humidity controlled at 30%RH.

2.3.2 Tensile tests

The Youngs modulus of the elastica loop is directly related to the compliance of the loop and therefore is a very important factor in the analysis. The Youngs modulus

of the elastomer was determined via tensile tests conducted as part of this study using an Instron 4505 universal testing frame. The tests were performed at room temperature under a constant crosshead rate of 5 mm/min using a pair of pneumatic grips to clamp the sample. Because the films are very thin (about 0.25 mm) and the modulus of the material is relatively low, attachment of an extensometer to the film would significantly affect the results of the measurement. To resolve this problem, the sample geometry was chosen to

21

be a rectangular strip instead of the regular dog-bone shape, and the strain was calculated based on the ratio of the crosshead displacement to the samples original length between grips. To reduce end effects, the ratio of the length to the width of the samples was

controlled to be greater than 10.

A typical stress-strain curve in the test from a particular batch of the PDMS material is shown in Figure 2.2. The Youngs modulus was obtained using a data fit

algorithm of the linear portion of the curve. Figure 2.3, Figure 2.4, and Figure 2.5 show the modulus data for specimens from different batches. These figures indicate that the stress-strain curves are very repeatable within a given batch but differ slightly from one batch to another. Table 2.1 summarizes all the modulus data for all the different batches

of the material tested. One possible reason for variations of the modulus among different batches is the lack of precise control of the ratio between the base material and the curing agent. As indicated by the data sheet provided by the Dow Corning Company, the content of the curing agent will have some effect on cure time and the physical properties

of the final cured elastomer. Lowering the curing agent concentration will result in a softer and weaker elastomer; increasing the concentration of the curing agent too much will result in over-hardening of the cured elastomer and will tend to degrade the physical and thermal properties.

2.3.3 Orientation of the elastica loop

A narrow strip was first cut from the cured PDMS film prepared earlier. The ends of the strips were lifted, bent, and fixed vertically with a known separation distance. The specimen was then attached to a sample holder as shown in Figure 2.6. The angles

of the PDMS strip at the fixed ends were 90. The height of the loop, h, depended on the

total length of the loop, 2L, and the distance between the clamped ends of the loop, 2C.

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2.4 Apparatus

There are four major components in the experimental apparatus as indicated in Figure 2.7: 1) the displacement control device; 2) the force measurement device; 3) the optical system; and 4) the computer control and data acquisition system (not shown in the figure). The elastica loop is directly attached to the shaft of the stepper motor, which is a major part of the displacement device and is controlled by a computer. When the elastica loop is brought into contact with a flat substrate by the displacement device, the interfacial attractive force produces a measurable change in contact area. The contact zones are observed in the monitor through a Nikon macro lens using a magnification

factor of 50x, and the contact length, 2B, can be measured directly from the monitor provided a careful calibration is performed before the test. At the same time, the contact force F between the loop and the flat substrate can be measured using an analytical balance mounted on the force measurement device. The whole contact sequence and data acquisition were controlled by the computer control and data acquisition system. In

the following sections, detailed descriptions for each component of the apparatus are given.

2.4.1 Displacement control device

Based on the dimensions of the specimen and the deformation level of the loop

induced by the interfacial interactions, a low speed, low vibration, and high-resolution displacement control system is required for the study. From a broad search of various positioning products, the IW-710 INCHWORM motor from Burleigh Instruments, Inc. was chosen as the major component for the high-resolution positioning system. This IW-710 INCHWORM motor uses compact piezoelectric ceramic actuators to achieve

nanometer-scale positioning steps over several millimeters. This motor offers a range of

motion for 6 mm and is featured with a mechanical resolution of approximately 4 nm over the entire range of motion with a maximum speed specified as 1.5 mm/sec. In

addition, this motor has a very high lateral stability with a lateral shaft runout of 0.2m.

23

This instrument also provides a non-rotating shaft and forward and reverse limit switches, providing a convenient way to control the contact procedure. The motor can sustain a light load (less than 1000 grams), and consequently, the elastica loop was directly attached to the shaft in the tests. The position of the elastica loop is read through an integral encoder associated with the IW-710 motor. The resolution of the encoder is

0.5 m with an accuracy up to 1 m. The motor and the encoder are controlled by the Burleigh 6200ULN closed-loop controller, which can be operated in either manual or computer control mode. In manual mode, the motor speed and the stop and start functions are all operated through the controller. In addition, the controller can actuate a

moving procedure consisting of a series of predetermined increments with the size being as small as the resolution of the encoder. If the controller is operating under the computer control mode, the motor is controlled by the controller through the 671 interface with the computer in a bi-directional communication mode. The bi-directional

interface has the following functions, which can be combined arbitrarily to perform a task: 1) load an absolute target displacement value; 2) load a step size; 3) set motor speed; 4) read current motor position; 5) read the current status of the motor and controller; 6) control position maintenance (On/Off); 7) perform motor stall test; 8) set zero reference and clear counter.

As the major component of the displacement device, the IW-710 motor is positioned in the apparatus as shown in Figure 2.8. The outer cylinder made of

polycarbonate [PC] rests on the base of the analytical balance and is used to support the motor and reduce air circulation around the elastica loop. The inner PC cylinder has no connection with the motor and rests on the scale pan of the balance. The substrate is placed on the top of the inner cylinder with the surface of interest facing down. When

the motor is controlled to move through the contact sequence, the elastica loop attached to the shaft is brought into contact with the substrate and then withdraws to separate the contact. The contact force is measured through the force measurement system, and simultaneously, the contact length is measured through the optical system.

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2.4.2 Force measurement device

The attractive force between two solid bodies in this study is several milligrams, which requires a high-resolution force measurement device. To satisfy this requirement, an SA 210 analytical balance from Scientech, Inc. was chosen. This balance is equipped

with a high standard of accuracy and has a resolution of 0.1 mg over the entire weight range of 210 grams. The balance is also equipped with a very reliable electronic filtering system, which helps to stabilize the weight reading when mechanical damping of the balance is insufficient. As a result, the display of the balance is prompt, clear, and

reliable. In addition, the stability indicator in the balance will also show an OK sign when the reading is valid insuring reliable results. In the setup for this study, the analytical balance is placed on a vibration reduction table along with the optical system, and is connected to the computer through an RS-232 interface for the purpose of data

acquisition.

2.4.3 Optical system and calibration

To measure the contact pattern and the contact length with adequate accuracy, an optical system is needed. In this study, the optical system mainly consists of a Nikon macro lens with up to 50X magnification, an FOI-150 fiber optic illuminator with two goosenecks, a TK-66 CCD Shutter camera from Micro-techica, Inc, a monitor, and a VCR as partially indicated in Figure 2.7. During the tests, the contact patterns are observed through the macro lens, and the FOI-150 provides uniform illumination in a focused area with white light. Through the digital video camera the entire contact

procedure can be observed and can be recorded if necessary using the VCR. The support of the macro lens also rests on the vibration reduction table to enhance the visual observation.

The contact length or contact area is measured directly from the monitor.

However, careful calibration is needed before any measurement is taken. To calibrate

the readings from the monitor, a fine scale with 20m resolution was set beside the

25

substrate underneath the micro lens and, once the system was in focus, readings of the scale were taken from the monitor. According to readings taken, a ruler was made and was directly attached to monitor, from which the contact length can be measured.

2.4.4 Computer control interface and data acquisition system

The computer control program used in this study was written in LabVIEW 5.0 from National Instruments [48]. LabVIEW is a graphic programming language, and is a very powerful tool in areas such as control, automated testing, and data acquisition. LabVIEW provides the flexibility and comprehensive functionality available in standard

C programming languages. A LabVIEW virtual instrument (VI) consists of a front panel and a block diagram. The source code uses an intuitive block diagram approach that works much like schematics and flow charts to solve problems. In addition, LabVIEW is platform-independent, so the programs created on one platform can easily be exported to

other platforms without any modification.

In this study, the control software written in LabVIEW 5.0 was used primarily to control the movement of the motor and collect the readings from the balance. The

analytical balance is equipped with an RS-232 interface and a subVI was therefore written to set up a bi-directional communication between the balance and the computer serial port in order to tare the balance and start or stop collecting data. The INCHWORM stepper motor is supplied with a data acquisition board and a control

subVI. Both the subVIs for the balance and for the motor were then imbedded into the main VI to form a complete control and data acquisition software. The final interface of the control software is shown in Figure 2.9.

2.5 Measurement using the JKR method

In order to compare and validate results obtained using the novel technique with those obtained using the JKR technique, measurements of the interfacial interactions using the JKR technique were taken using the apparatus discussed above. First,

26

hemispherical lenses and thin films using the SYLGARD 184 silicone elastomer (PDMS) were prepared. The hemispherical lenses were provided by Lehigh University and the thin films were then coated on glass substrates. The hemispherical lens was directly

attached on top of the shaft of the step motor. A glass substrate was then set on the inner cylinder of the apparatus with the side of the surface coated with the same PDMS material facing down as shown in Figure 2.10.

When the test began, as controlled by the computer, the PDMS lens moved upward slowly until the lens was very close to the substrate, where the interfacial interactions are high enough and the surface of the lens jumped forward to be in contact with the substrate. Hence, a finite circular contact area resulted. The contact

area increased as the stepper motor continued to move upward and then decreased as the unloading cycle started. The whole contact procedure and the contact pattern were recorded through the digital video camera and the VCR, and the bi-directional data acquisition system allowed the contact force to be monitored instantaneously. As the

unloading continued, the contact force decreased. When the contact force decreased to zero, a finite contact still remained at the interface, which was clearly observed during all the tests. Consequently, a tensile force was required to separate the lens and the substrate, a well-known feature of the JKR technique.

Figure 2.11 shows a typical contact pattern resulting from the contact between the lens and the flat glass substrate coated with the PDMS. The result of the contact radius versus the contact force for both the loading and unloading cycle is shown in Figure 2.12. The positive values represent compressive contact forces between the PDMS lens and the

substrate, and the negative values represent tensile forces, which were required to separate the contact. The data indicate that there is a small hysteresis between the loading and unloading cycles, similar to what She et al. [35] observed for similar material systems.

27

The data were then analyzed using the JKR theory and a numerical regression method as discussed in Appendix A was used to obtain Wadh and K. Figure 2.13 shows the fitting curves of the experimental data, where a3 is plotted as a function of P. The

calculation method is based on the work in reference [14]. The best fit between the experimental loading data and equation (9) yielded values Wcoh = 45.8 mJ/m2 and K = 1.73 MPa, which are consistent with the values measured by other researchers [[14], [17]]. However, the best fit between the experimental unloading data and equation (9) yielded Wcoh = 60.2 mJ/m2. The resultant high value for Wcoh in unloading procedure is possibly due to energy loss or molecular interdiffusion across the interface during the loading and unloading cycle, which has not been taken into account in the JKR theory. The data from the loading cycle were therefore chosen to determine the surface and

interfacial energies in this study. According to equation (1), the surface energy of the PDMS material is half of the work of cohesion in this case and thus = 22.9 mJ/m2.

2.6 Measurement using elastica loop

In this test, the PDMS films coated on glass substrates were prepared in the same

way as discussed in the previous section. The PDMS elastica loop was carefully attached to the shaft of the motor with a tight connection. Before the test started, a volume static eliminator VSE 3000 from Chapman Corp. was used to remove static charges through blowing both the loop and the substrate for 10 minutes. Following a similar control

algorithm as in the JKR test, the loop moved upward and then contact occurred between the loop and the substrate as shown in Figure 2.14. The contact length increased as the loading cycle continued and then decreased when the unloading cycle started. Finally, separation occurred when a tensile force was applied on the loop. Considering the

effects of the interfacial interactions acting on the system and the time needed to reach

equilibrium, the loading speed was selected as 10.1 m/sec for each step, and between any two steps during the loading cycle, the sample was allowed to dwell for an interval of 200 sec. For the unloading cycle, the dwelling interval was increased to 300 sec/step, because the thermodynamic equilibrium took more time to reach. The whole contact

28

procedure as well as the contact patterns and contact forces were again recorded and stored for future analysis.

2.6.1 Contact patterns

Similar to the situation in the JKR test, when the elastica loop was close enough

to the substrate, the loop jumped forward to form a contact with the substrate because of the interfacial energy. However, in this case, due to the anticlastic bending caused by the Poissons effect, a transverse curvature is resulted when the loop is bent as shown in Figure 2.15. Consequently, the contact first happened at the two edges of the loop as indicated in Figure 2.16. As the shaft continue to move toward the substrate, the contact zone spread across the width and propagated outward in the longitudinal direction toward the loop ends. However, the contact front still remained curved because of the anticlastic bending as shown in Figure 2.17. As the contact area continued to grow, the curved

contact front is self-similar as shown by Figure 2.18, which represents a typical contact pattern resulting from the contact between a PDMS elastica loop and a flat substrate coated with the same material. Because of the arched contact front, the determination of the contact length was very difficult. In this study, as indicated in Figure 2.18, the

contact length is either defined as the length of the longitudinal center line of the contact zone 2B, which is also the lower bound of the contact length, or the length between the outer edges of the contact fronts 2Bo, which in fact is the upper bound of the contact length. When 2B>>2Bo, the contact area at corner was no longer negligible. During the

unloading cycle, a finite contact area remained even when the contact force was zero, and a tensile force was required to separate the contact. Because of the compliant structure, the contact force, F, detected in this test were in general smaller than those in the JKR experiment, but the contact lengths, 2B, were normally larger than the contact diameter,

2a, observed in the JKR test.

29

2.6.2 Experimental results and discussion

In the following, detailed experimental observations and discussions are given for the PDMS elastica loop in contact with different substrates.

1. PDMS loop to PDMS substrate

In this test, both the elastica loop and substrate surface were made of the PDMS elastomer. The substrate consists of a piece of glass and a layer of the PDMS film about 0.2 mm thick. The film was coated using the same method discussed earlier. The glass

was cleaned with acetone and de-ionized water before coating. The dimensions of the

elastica strip were 14.7 mm 0.96 mm 0.17 mm. The distance between the two ends

of the loop, 2C, was 6.38 mm and the modulus of the loop was 1.81 MPa.

A plot of contact length versus the contact force was obtained in the test and is

shown in Figure 2.19. In the figure, a negative force value indicates a tensile force between the elastica loop and the substrate, and a positive force value indicates a compressive force. The diamond and square data are the loading and unloading curves corresponding to the inner contact lengths, 2B, which are the lower bounds of the contact

length; and the triangle and circle data are the loading and unloading curves corresponding to the outer contact lengths, 2Bo, which are the upper bounds of the contact length. Figure 2.19 indicates that the geometry of the contact front does not change during the test because the difference between the inner contact curve and outer contact curve remains almost the same. More specifically, in the test,

2Bo=2B+0.350.04 mm (10) The result indicates that the contact spread occurred in a self-similar manner. Because the analytical solution, which will be discussed later, was developed based on beam theory and the effect of anticlastic bending was ignored, the choice of location to determine the contact length will influence the comparison between the experimental

results and analytical predictions.

30

Both the unloading curves in Figure 2.19 indicate that a finite contact area still existed even when the contact force was zero and a tensile force was required to separate

the contact.

Figure 2.19 also indicates that at the beginning of the loading cycle, the forces increased with the displacement at a relatively low rate. When the contact length

increased to about 40% of the loop width (specified by the dashed line), the slope of the curve increased, suggesting that the contact area started to spread at a relatively high rate. The occurrence of this contact spreading transition is due to the effects of the main bending of the loop in longitudinal direction and the anticlastic bending in transverse

direction. As the contact started, the loop first needed to overcome the anticlastic bending curvature. When the contact length exceeded about 40% of the loop width, the loop was flattened, the anticlastic bending effect was already eliminated, and the compliance of the loop substantially increased. In fact, the transition was difficult to

capture but could be easily observed in unloading procedure, so the transition was determined using the unloading data. Consequently, the contact area started to spread at a relatively high rate. The contact forces detected in this test were in general smaller than those observed in the JKR test and the contact lengths (both the lower and upper bounds) were larger than the contact diameter observed in the JKR test because of the compliant structure of the loop. A plot of the contact force versus the displacement of the loop of this test is shown in Figure 2.20, which indicates that a small hysteresis loop exists in the loading and unloading curve, which will be discussed later.

2. PDMS loop to glass substrate

In this test, the substrate was changed to a piece of glass cleaned with acetone and

de-ionized water. The dimensions of the elastica loop were 15.8 mm 0.94 mm 0.17

mm with a modulus of 1.81 MPa, and the distance between the two ends of the loop was 6.38 mm. A plot of contact length versus contact force is shown in Figure 2.21, and a

31

plot of contact force versus the displacement of the loop is shown in Figure 2.22. As same as the previous experiment, the spread transition was clarified using the unloading data. Figure 2.21 shows that in the beginning of the loading cycle, the forces increased

slowly with the contact length until the contact length was about one-third of the loop width, where the slope of the curve increased and the contact started to spread rapidly. As compared to the case of PDMS loop in contact with PDMS substrate, the loading and unloading curve shapes are very different, i.e., the hysteresis increased with the

decreased unloading force, which indicats a very different interfacial interaction between the PDMS loop and glass substrate. The reason may be stronger interdiffusion between the two materials or larger attractive force between the molecules on the two contact surfaces was occurred. Small hysteresis was observed in the contact force versus the

displacement of the loop curve as shown in Figure 2.22.

3. PDMS loop to polycarbonate [PC] plate

In this test, a PC plate cleaned with soap water and DI water was selected as the substrate, and the surface of the PC plate is very smooth. The dimensions of the elastica

strip were 15.3 mm 0.61 mm 0.21 mm, with a modulus of 2.00 MPa, and the distance

between the two ends of the loop was 6.55 mm. A plot of contact length versus contact force and a plot of contact force versus displacement of the loop are shown in Figure 2.23 and Figure 2.24, respectively, similar to the experiments of PDMS loop in contact with the glass substrate, the contact spread transition in this case occurred when the contact length was about half of the loop width, and hysteresis existed between the

loading and unloading curves.

4. PDMS loop to commercial cellulose acetate substrate

Following the same contact procedure, the measurement between the PDMS

elastica loop and a commercial cellulose acetate tape (3M ScotchTM Transparent Tape 600) was taken. The tape was adhered to a glass substrate to conduct the test. The

32

dimensions of the elastica strip were 16.8 mm 0.84 mm 0.17 mm with a modulus of

1.81 MPa, and the distance between the two ends of the loop was 6.38 mm. A plot of contact length versus contact force and of contact force versus the displacement of the loop are shown in Figure 2.25 and Figure 2.26, respectively, the contact spread transition in this test was observed when the contact length was about half of the loop width. A hysteresis loop was also observed.

2.6.3 Hysteresis

In the elastica loop tests, a large hysteresis with a tail was observed between the loading and the unloading curve for every test. To investigate the hysteresis, the creep

and the hysteresis properties of the PDMS elastomeric material were characterized. Creep tests at room temperature were conducted on a TA Instruments Dynamic Mechanical Analyzer (DMA) 2980, and the creep time used was 8 hours, which corresponds to the longest running time ever experienced during the elastica loop tests.

The hysteresis was characterized using tensile tests conducted on the screw-driven Instron 4505 machine and the tensile hysteresis tests were conducted for about three loading and unloading cycles at a rate of 5mm/min with maximum strain around 40%.

A typical creep test result is shown in Figure 2.27. The results indicate that only a negligible amount of creep occurred. The hysteresis tensile test data are shown in Figure 2.28. This figure shows that the PDMS material has a very low hysteresis since the loading and unloading curves were almost on top of each other.

The creep and hysteresis test results indicate that the viscoelastic properties of the PDMS material give a very limited contribution to the hysteresis observed in the contact of the PDMS elastica loop with various substrates. The hysteresis is believed to be primarily due to the interfacial interactions between the two surfaces at the interface.

33

To further investigate the hysteresis problem, the PDMS elastica loop versus PDMS substrate contact tests were conducted again. However, in this experiment, the loading and unloading cycles were performed in a cyclic fashion for three cycles. The

loading rate was still 10.1 m/sec for each step, and between any two steps during the loading cycle, the sample was allowed to dwell for an interval of 200 sec. For the unloading cycle, the dwelling interval was increased to 300 sec/step, because it took more time to reach the thermodynamic equilibrium. The experimental results for contact radius versus contact force in Figure 2.29 clearly show that the hysteresis increased significantly with the contact cycle since the area enclosed by the loading and unloading

curves increased from the first contact cycle to the third contact cycle. This increase of hysteresis with the contact cycle is possibly due to the molecular inter-diffusion between the two surfaces occurred when they were in contact, which altered the interfacial interactions between these two surfaces.

The effect of contact rate on the contact hysteresis was also investigated in this study using the same testing procedure. In the tests, the loading and unloading cycles were performed continuously using constant testing rates with zero dwell time between

any two steps, and two testing rates were evaluated in this study.

Plots of contact radius versus contact force obtained in the tests are shown in

Figure 2.30. A testing rate of 5.1 m/sec was the highest rate that could be achieved with the current apparatus. Figure 2.30 shows that although not significant, the contact

hysteresis increases with the testing rate, which again indicates that the interfacial interaction is a time-dependent phenomenon.

34

Table 2.1 Youngs modulus of the tensile specimens from different batches.

Batch No. 1 2 3 4 5 Number of Samples 8 3 3 4 4

Modulus (MPa) 2.28 1.55 1.64 2.00 2.50 Standard Deviation

(MPa) 0.06 0.14 0.03 0.07 0.08

35

Figure 2.1 Elastica loop configuration

%

&

)

36

\ [

5

6WUDLQ

6WUHVV

03D

Figure 2.2 A typical stress-strain plot for a PDMS strip.

37

Figure 2.3 Youngs moduli for eight PDMS samples of the first batch and

their average. The error bar represents one standard

deviation.

0 significantly but agree with the analytical solution with = 0 more. As discussed earlier, according to the analytical solution [45], when the loop is in contact with the substrate, the contact pressure in the contact region is zero. Since the

surface attractive force only acts within a very small range near the surface and decreases rapidly with the distance from the surface (proportional to r-3 according to reference 3, where the r is the distance from the surface). Consequently, a small asperity on the surface of the loop or slight contamination on the substrate may decrease the degree of

the intimate contact between these two surfaces in the center of the loop. However, this decrease in the degree of intimate contact may not be able to observe optically under the current microscope system because the wavelengths used might be essentially larger than

102

the distance between these two surfaces in the center of the loop. Although in the tests of this study, the PDMS films were all carefully prepared and cured with little control of the surface smoothness, microscopic-scale asperity or blisters caused by a small amount of

air trapped in the PDMS gel may still exist, which would certainly decrease the degree of intimate contact between the two surfaces.

The testing apparatus may also have possible sources of errors. In the analytical

solution, the vertically symmetric line of the loop passing through the center of the loop

and the middle point of the end separation distance were assumed to be perfectly perpendicular to the substrate. However, this perfect alignment was very difficult to accomplish during the tests, which may cause slight changes in the readings of the

contact lengths. Additionally, in the analytical solution, the substrate was considered as a flat, rigid, and smooth surface. But in the experiments, the coatings with certain thickness were not perfectly rigid, which, although the effect was estimated to be small in this study, may still cause some errors for the tests. Knowing these possible sources of

error will help us improve the accuracy of the analytical solution and the experimental measurements.

103

Table 4.1 Nondimensionalization process Nondimensional Quantity Definition

Arc length, s S/L

Horizontal coordinate, x X/L

Vertical coordinate, y Y/L

Height of the loop, h H/L

Contact length, b B/L

Distance between the ends, c C/L

Displacement, /L Horizontal force, p FL2/(EI)

Contact force, q QL2/(EI) Moment at clamped end, mo Mo L/(EI)

Moment at beginning of contact, mb Mb L/(EI) Work of adhesion, wadh Wadh L2/(EI)

Energy, u(E, M, A, or T) U(E,M,A, or T) L/(EI)

104

Figure 4.1 An elastica loop in contact with a flat, rigid, smooth, horizontal surface. After reference 46.

E

F

T

V

105

Figure 4.2 Definitions of variables. After reference 46.

+

P

R

T

T

S

S

P

R [

\

V

106

Figure 4.3 Model adhesion effects. After reference 46.

P

E

P

E

107